Given a set of n real numbers with a nonnegative sum, consider the family of all its k-element subsets with nonnegative sums. How small can the size of this family be? We show that this problem is closely related to a problem raised by Ahlswede and Khachatrian in . The latter, in a special case, is nothing else but the problem of determining a minimal number c n (k) such that any k-uniform hypergraph on n vertices having c n (k) + 1 edges has a perfect fractional matching. We show that results obtained in  can be applied for the former problem. Moreover, we conjecture that these problems have in general the same solution.
Problems of Information Transmission – Springer Journals
Published: Jan 24, 2013
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